From Stdlib Require Import Classical Relations.

Section Pilot.
(* Vocabulary.  Nothing is assumed about any of these. *)
Variable Thing : Type.
Variable Gives : Thing -> Thing -> Prop.     (* Gives y x : y gives existence to x *)
Variable Through : Thing -> Thing -> Prop.   (* Through x y : x has its existence through y *)
Variable Exists : Thing -> Prop.             (* x exists *)
Variable Ess : Thing -> Prop.                (* x's existence is from its own nature *)
Variable PartOf : Thing -> Thing -> Prop.    (* p is a part of u *)
Variable Universe : Thing.                   (* the universe *)

(* Definitions. *)
Definition DependsOn (x y : Thing) : Prop := Gives y x.            (* x depends on y *)
Definition ThroughAnother (x : Thing) : Prop := exists y, Gives y x.
Definition OfItself (x : Thing) : Prop := Exists x /\ ~ ThroughAnother x.
Definition Necessary (x : Thing) : Prop := Exists x /\ Ess x.
Definition Possible (x : Thing) : Prop := Exists x /\ ~ Ess x.
Definition InUniverse (x : Thing) : Prop := x = Universe \/ PartOf x Universe.

(* Approved postulates, written as definitions: a theorem uses one only when
   its statement lists it as a condition. *)
Definition MP001 : Prop := forall x y, Gives y x -> ~ Through y x.   (* a giver does not have its existence through what it gives existence to *)
Definition MP002 : Prop := forall x y, Gives y x -> Through x y.     (* the receiver has its existence through the giver *)
Definition MP003 : Prop := forall x y z, Through x y -> Through y z -> Through x z.  (* existence-through is inherited *)
Definition MP008 : Prop := forall x, Exists x -> ~ ThroughAnother x -> Ess x.       (* existence comes from one's nature or from a giver *)
Definition OP001 : Prop := forall x y, Gives y x -> OfItself y.      (* only what exists of itself gives existence *)
Definition OP006 : Prop := exists x, InUniverse x /\ Possible x.     (* the universe is not wholly necessary *)
Definition SP001 : Prop := exists x, Exists x.                       (* something exists *)

(* Added for the full test: aggregates, possible worlds, and their postulates. *)
Variable MemberOf : Thing -> Thing -> Prop.  (* p is a member of the aggregate a *)
Variable World : Type.
Variable ExistsIn : World -> Thing -> Prop.  (* x exists in world w *)
Definition Independent (x : Thing) : Prop := forall y, ~ DependsOn x y.
Definition NonExistencePossible (x : Thing) : Prop := exists w, ~ ExistsIn w x.
Definition AggregateOfPossibles (A : Thing) : Prop :=
  (forall p, p <> A -> (MemberOf p A <-> Possible p)) /\ ~ MemberOf A A.
Definition MP007 : Prop := forall x, Ess x -> forall w, ExistsIn w x.        (* existence from one's nature cannot fail *)
Definition MP009 : Prop := (exists p, Possible p) -> exists A, AggregateOfPossibles A /\ Exists A.  (* the aggregate of possibles exists *)
Definition MP010 : Prop := forall A, (exists p, MemberOf p A) ->
  (forall p, MemberOf p A -> ~ Ess p) -> ~ Ess A.                            (* possibles do not add up to a necessary *)
Definition MP011 : Prop := forall c A p, Gives c A -> MemberOf p A -> clos_trans Thing Gives c p.  (* giving to a whole gives to its members *)
Definition MP012 : Prop := forall y x, Gives y x -> Exists y.               (* a giver exists *)

Lemma imkan_s8_no_internal_cause_L5 : MP001 -> MP002 -> MP003 -> MP010 -> MP011 -> forall A, AggregateOfPossibles A -> (forall x, ~ Gives x x) /\ ~ Gives A A /\ (forall c, MemberOf c A -> ~ Gives c A) /\ (forall B, B <> A -> Exists B -> (exists p, MemberOf p B) -> (forall p, MemberOf p B -> MemberOf p A) -> ~ Gives B A).
Proof.
intros H1 H2 H3 H10 H11 A [HA HAA].
assert (g: forall x, ~ Gives x x).
{ intros x h. exact (H1 x x h (H2 x x h)). }
assert (ct: forall a b, clos_trans Thing Gives a b -> Through b a).
{ intros a b h. induction h as [a b h|a b c h1 IH1 h2 IH2].
  - exact (H2 b a h).
  - exact (H3 c b a IH2 IH1). }
assert (m: forall c, MemberOf c A -> ~ Gives c A).
{ intros c hc hg.
  pose proof (H11 c A c hg hc) as h.
  apply clos_trans_t1n in h.
  inversion h; subst;
  [ exact (g _ H) | exact (H1 _ _ H (ct _ _ (clos_t1n_trans _ _ _ _ H0))) ]. }
repeat split.
- exact g.
- exact (g A).
- exact m.
- intros B hne hE hex hsub hg.
  assert (hP: Possible B).
  { split; [exact hE|].
    apply (H10 B hex).
    intros p hp.
    assert (hpA := hsub p hp).
    assert (pne: p <> A) by (intro e; subst; exact (HAA hpA)).
    pose proof (proj1 (HA p pne) hpA) as [_ ne].
    exact ne. }
  exact (m B (proj2 (HA B hne) hP) hg).
Qed.

End Pilot.

Print Assumptions imkan_s8_no_internal_cause_L5.